A short note on the relation between the optical theorem and the Pauli exclusion principle

Using the optical theorem, we focus on a specific class of forward-scattering diagrams with imaginary kinematics that describe processes in which two identical fermions appear to occupy the same state. What initially seems to be in conflict with the Pauli exclusion principle turns out to be a key element of how the principle manifests itself in scattering theory.

💡 Research Summary

The paper investigates a subtle interplay between the optical theorem—a direct consequence of S‑matrix unitarity—and the Pauli exclusion principle, focusing on a class of forward‑scattering diagrams that contain crossed fermion lines. Starting from the standard optical theorem relation
(2,\text{Im},T_{ii}= \sum_f |T_{fi}|^2),
the author notes that for individual diagrams with an on‑shell cut one can identify a contribution to the total transition probability. Three typical cut topologies are illustrated (Fig. 1): a conventional self‑energy cut, an anomalous‑threshold cut that appears only when an unstable particle is involved, and a “crossed‑leg” cut that separates the diagram into two disconnected pieces. The work concentrates on the third case, where the cut separates two fermionic legs.

A toy model is introduced: two Majorana fermions (\chi_1) (mass (m_1)) and (\chi_2) (mass (m_2)) interact via a light neutral scalar (\phi) (mass (m_\phi)) through the Yukawa‑type Lagrangian
(\mathcal L = -g,\bar\chi_2\chi_1\phi).
Assuming (m_2 > m_1 + m_\phi), the decay (\chi_2 \to \chi_1\phi) is kinematically allowed. The author imagines a collider where a beam of (\chi_1) particles strikes a target of (\chi_2) particles. The lowest‑order forward‑scattering diagram (Fig. 2a) consists of two vertices and a scalar propagator that can be cut, yielding a final state (\chi_1 + \chi_1 + \phi). At first glance this appears to violate the Pauli principle because the two (\chi_1) fermions would occupy the same momentum and spin state.

To resolve the apparent paradox, the paper explicitly constructs the two possible amplitudes that lead to the same final state. The first amplitude (M_1) contracts the incoming (\chi_1) with the first outgoing (\chi_1); the second amplitude (M_2) contracts it with the second outgoing (\chi_1). Because fermion operators anticommute, (M_2) acquires an overall minus sign relative to (M_1). Both amplitudes contain delta‑functions enforcing momentum conservation and Kronecker deltas for spin matching. The transition probability is then (|M_1+M_2|^2), which expands to three terms: (|M_1|^2), (|M_2|^2) (each giving the ordinary decay width of (\chi_2\to\chi_1\phi)), and the interference term (2\text{Re}(M_1M_2^*)).

Carrying out the phase‑space integrals shows that the sum of the first two terms reproduces the standard differential width (d\Gamma) (Eq. 12). The interference term, however, is non‑zero only when the momentum and spin of the produced (\chi_1) exactly match those of the incoming beam particle. In that special kinematic configuration the interference contributes a term (-d\Gamma) (Eq. 14), which precisely cancels the ordinary width. Consequently, the net contribution of configurations where two identical fermions would occupy the same quantum state vanishes, fully respecting the Pauli exclusion principle.

The paper emphasizes several key insights:

1. Crossed‑leg forward‑scattering diagrams cannot be interpreted as isolated physical processes; they must be paired with the corresponding disconnected amplitude to form a complete, unitary contribution.
2. The antisymmetry of fermionic operators automatically generates the minus sign that enforces Pauli blocking through interference.
3. This mechanism is intimately related to anomalous thresholds and t‑channel singularities; even when such singularities appear, unitarity and fermionic statistics conspire to produce a physically sensible result.

In the concluding section the author reiterates that the optical theorem remains perfectly compatible with the Pauli principle. The apparent violation is only superficial, arising from an incomplete treatment of the cut diagram. When the full set of interfering amplitudes is taken into account, contributions from identical‑state fermion pairs cancel exactly, preserving both unitarity and the exclusion principle. This analysis clarifies a subtle point in perturbative quantum field theory and provides a template for handling similar crossed‑leg contributions in more complex theories or kinetic equations.